memoire-m2
My M2 Memoire on mapping class groups & their representations
representations.tex (26577B)
1 \chapter{Low-Dimensional Representations}\label{ch:representations} 2 3 Having built a solid understanding of the combinatorics of Dehn twists, we are 4 now ready to attack the problem of classifying the representations of 5 \(\Mod(\Sigma_g)\) of sufficiently small dimension. As promised, our strategy 6 is to make use of the \emph{geometrically-motivated} relations derived in 7 Chapter~\ref{ch:dehn-twists} and Chapter~\ref{ch:relations}. 8 9 Historically, these relations have been exploited by Funar \cite{funar}, 10 Franks-Handel \cite{franks-handel} and others to establish the triviality of 11 low-dimensional representations, culminating in Korkmaz' \cite{korkmaz} recent 12 classification of representations of dimension \(n \le 2 g\) for \(g \ge 3\). 13 The goal of this chapter is to provide a concise account of Korkmaz' results. 14 15 \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} 16 Let \(\Sigma_g^b\) be the compact surface of genus \(g \ge 1\) with \(b\) 17 boundary components and \(\rho : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\) be 18 a linear representation with \(n < 2 g\). Then the image of \(\rho\) is 19 Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial. 20 \end{theorem} 21 22 Like some of the results we have encountered so far, the proof of 23 Theorem~\ref{thm:low-dim-reps-are-trivial} is elementary in nature: we proceed 24 by induction on \(g\) and tedious case analysis. We begin by the base case \(g 25 = 2\). 26 27 \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case} 28 Given \(\rho : \Mod(\Sigma_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the 29 image of \(\rho\) is Abelian. 30 \end{proposition} 31 32 \begin{proof}[Sketch of proof] 33 Given \(\alpha \subset \Sigma_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and 34 denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v = 35 \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2, 36 \gamma, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_2^b\) be the curves of the 37 Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in 38 Figure~\ref{fig:lickorish-gens-genus-2}. 39 \begin{figure} 40 \centering 41 \includegraphics[width=.2\linewidth]{images/lickorish-gens-gen-2.eps} 42 \caption{The Lickorish generators for $g = 2$.} 43 \label{fig:lickorish-gens-genus-2} 44 \end{figure} 45 46 If \(n = 1\) then \(\rho(\Mod(\Sigma_2^b)) \subset \GL_1(\mathbb{C}) = 47 \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by 48 Proposition~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1} 49 = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker 50 \rho\) and thus \(\Mod(\Sigma_2^b)' \subset \ker \rho\) -- i.e. 51 \(\rho(\Mod(\Sigma_2^b))\) is Abelian. Given the braid relation 52 \begin{equation}\label{eq:braid-rel-induction-basis} 53 L_{\alpha_1} L_{\beta_1} L_{\alpha_1} 54 = L_{\beta_1} L_{\alpha_1} L_{\beta_1}, 55 \end{equation} 56 this amounts to showing \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute. 57 58 To that end, we exhaustively analyze all of the possible Jordan forms 59 \begin{align*} 60 \begin{pmatrix} 61 \lambda & 0 \\ 62 0 & \lambda 63 \end{pmatrix} 64 & \quad{\normalfont(1)} 65 & 66 \begin{pmatrix} 67 \lambda & 0 \\ 68 0 & \mu 69 \end{pmatrix} 70 & \quad{\normalfont(2)} 71 & 72 \begin{pmatrix} 73 \lambda & 1 \\ 74 0 & \lambda 75 \end{pmatrix} 76 & \quad{\normalfont(3)} 77 \\ 78 \begin{pmatrix} 79 \lambda & 0 & 0 \\ 80 0 & \lambda & 0 \\ 81 0 & 0 & \lambda 82 \end{pmatrix} 83 & \quad{\normalfont(4)} 84 & 85 \begin{pmatrix} 86 \lambda & 0 & 0 \\ 87 0 & \mu & 0 \\ 88 0 & 0 & \nu 89 \end{pmatrix} 90 & \quad{\normalfont(5)} 91 & 92 \begin{pmatrix} 93 \lambda & 1 & 0 \\ 94 0 & \lambda & 1 \\ 95 0 & 0 & \lambda 96 \end{pmatrix} 97 & \quad{\normalfont(6)} 98 \\ 99 \begin{pmatrix} 100 \lambda & 0 & 0 \\ 101 0 & \mu & 1 \\ 102 0 & 0 & \mu 103 \end{pmatrix} 104 & \quad{\normalfont(7)} 105 & 106 \begin{pmatrix} 107 \lambda & 0 & 0 \\ 108 0 & \lambda & 1 \\ 109 0 & 0 & \lambda 110 \end{pmatrix} 111 & \quad{\normalfont(8)} 112 & 113 \begin{pmatrix} 114 \lambda & 0 & 0 \\ 115 0 & \lambda & 0 \\ 116 0 & 0 & \mu 117 \end{pmatrix} 118 & \quad{\normalfont(9)} 119 \end{align*} 120 of \(L_{\alpha_2}\) -- where \(\lambda, \mu, \nu \in \mathbb{C}^\times\) are 121 all distinct. By changing basis we may assume without loss of generality that 122 the matrix \(L_{\alpha_2}\) is exactly its Jordan form, so that \(E_{\alpha_2 123 = \lambda} = \bigoplus_{i \le \dim E_{\alpha_2}} \mathbb{C} e_i\). 124 125 For cases (1) to (6), we use the change of coordinates principle and 126 different relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) lie inside 127 some Abelian subgroup of \(\GL_n(\mathbb{C})\). 128 129 \begin{enumerate}[leftmargin=1.9cm] 130 \item[\bfseries\color{highlight}(1) \& (4)] 131 By the change of coordinates principle, both \(L_{\alpha_1}\) and 132 \(L_{\beta_1}\) are conjugate to \(L_{\alpha_2} = \lambda\). But the 133 only matrix conjugate to \(\lambda\) is \(\lambda\) itself. Hence 134 \(L_{\alpha_1} = L_{\beta_1} = \lambda \in \mathbb{C}^\times\). 135 136 \item[\bfseries\color{highlight}(2) \& (5)] 137 Since \(\alpha_2\) is disjoint from both \(\alpha_1\) and \(\beta_1\), it 138 follows from the disjointness relations \([\tau_{\alpha_1}, 139 \tau_{\alpha_2}] = [\tau_{\beta_1}, \tau_{\alpha_2}] = 1\) that 140 \(L_{\alpha_1}\) and \(L_{\beta_1}\) preserve the eigenspaces of 141 \(L_{\alpha_2}\), which are all \(1\)-dimensional. Hence \(L_{\alpha_1}\) 142 and \(L_{\beta_1}\) lie inside the subgroup of diagonal matrices -- an 143 Abelian subgroup of \(\GL_n(\mathbb{C})\). 144 145 \item[\bfseries\color{highlight}(3) \& (6)] 146 As before, it follows from the disjointness relations that \(E_{\alpha_2 147 = \lambda} = \ker (L_{\alpha_2} - \lambda)\) and \(\ker (L_{\alpha_2} - 148 \lambda)^2\) are invariant under both \(L_{\alpha_1}\) and 149 \(L_{\beta_1}\). This implies \(L_{\alpha_1}\) and \(L_{\beta_1}\) are 150 upper triangular matrices with \(\lambda\) along their diagonals. Any 151 such pair of matrices satisfying the braid relation 152 (\ref{eq:braid-rel-induction-basis}) commute. 153 \end{enumerate} 154 155 Similarly, in case (7) we use the braid relation and the disjointness 156 relations to show \(L_{\alpha_1}\) and \(L_{\beta_1}\) commute -- see 157 \cite[Proposition~5.1]{korkmaz} for a full proof. Cases (8) and (9) require 158 some extra thought. We consider the curve \(\beta_2\). In these cases, the 159 eigenspace \(E_{\alpha_2 = \lambda}\) is \(2\)-dimensional. Since 160 \(L_{\alpha_2}\) and \(L_{\beta_2}\) are conjugate, \(E_{\beta_2 = \lambda}\) 161 is also \(2\)-dimensional -- indeed, conjugate operators have the same Jordan 162 form. Now either \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) or 163 \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\). We begin by the first 164 case. 165 166 We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then 167 \(E_{\alpha_2 = \lambda}\) is \(\Mod(\Sigma_2^b)\)-invariant. Indeed, by 168 Observation~\ref{ex:change-of-coordinates-crossing} we can always find \(f, 169 g, h_i \in \Mod(\Sigma_2^b)\) with 170 \begin{align*} 171 f \cdot [\alpha_2] & = [\alpha_1] 172 & 173 g \cdot [\alpha_2] & = [\beta_1] 174 & 175 h_i \cdot [\alpha_2] & = [\alpha_2] \\ 176 f \cdot [\beta_2] & = [\beta_1] 177 & 178 g \cdot [\beta_2] & = [\gamma] 179 & 180 h_i \cdot [\beta_2] & = [\eta_i]. 181 \end{align*} 182 In particular, 183 \begin{align*} 184 f \tau_{\alpha_2} f^{-1} & = \tau_{\alpha_1} 185 & 186 g \tau_{\alpha_2} g^{-1} & = \tau_{\beta_1} 187 & 188 h_i \tau_{\alpha_2} h_i^{-1} & = \tau_{\alpha_2} \\ 189 f \tau_{\beta_2} f^{-1} & = \tau_{\beta_1} 190 & 191 g \tau_{\beta_2} g^{-1} & = \tau_{\gamma} 192 & 193 h_i \tau_{\beta_2} h_i^{-1} & = \tau_{\eta_i}. 194 \end{align*} 195 and thus 196 \begin{align*} 197 E_{\alpha_1 = \lambda} 198 = \rho(f) E_{\alpha_2 = \lambda} 199 & = \rho(f) E_{\beta_2 = \lambda} 200 = E_{\beta_1 = \lambda} 201 \\ 202 E_{\beta_1 = \lambda} 203 = \rho(g) E_{\alpha_2 = \lambda} 204 & = \rho(g) E_{\beta_2 = \lambda} 205 = E_{\gamma = \lambda} 206 \\ 207 E_{\eta_i = \lambda} 208 = \rho(h_i) E_{\alpha_2 = \lambda} 209 & = \rho(h_i) E_{\beta_2 = \lambda} 210 = E_{\beta_2 = \lambda}. 211 \end{align*} 212 In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} = 213 E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} = 214 E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} = \lambda}\) is invariant 215 under the action of all Lickorish generators. 216 217 Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : 218 \Mod(\Sigma_2^b) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- 219 recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By 220 case (2), \(\bar \rho(f) = 1\) for all \(f \in \Mod(\Sigma_2^b)'\), given 221 that \(\bar \rho(\Mod(\Sigma_2^b))\) is Abelian. Thus 222 \[ 223 \rho(\Mod(\Sigma_2^b)') \subset 224 \begin{pmatrix} 225 1 & 0 & * \\ 226 0 & 1 & * \\ 227 0 & 0 & * 228 \end{pmatrix} 229 \] 230 lies inside the group of upper triangular matrices, a solvable subgroup of 231 \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we 232 get \(\rho(\Mod(\Sigma_2^b)') = 1\): any homomorphism from a perfect group to 233 a solvable group is trivial. 234 235 Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and 236 the Jordan form of \(L_{\alpha_2}\) is given by (8) then the disjointness 237 relations \([\tau_{\alpha_2}, \tau_{\alpha_1}] = [\tau_{\alpha_2}, 238 \tau_{\beta_1}] = [\tau_{\beta_2}, \tau_{\alpha_1}] = [\tau_{\beta_2}, 239 \tau_{\beta_1}] = 1\) implies that \(L_{\alpha_1}\) and \(L_{\beta_1}\) 240 preserve the eigenspaces of both \(L_{\alpha_2}\) and \(L_{\beta_2}\), so 241 \[ 242 0 243 \subsetneq E_{\alpha_2 = \lambda} \cap E_{\beta_2 = \lambda} 244 \subsetneq E_{\alpha_2 = \lambda} 245 \subsetneq V 246 \] 247 is a flag of subspaces invariant under \(L_{\alpha_1}\) and \(L_{\beta_1}\). 248 In this case we can find a basis for \(\mathbb{C}^3\) with respect to which 249 the matrices of \(L_{\alpha_1}\) and \(L_{\beta_1}\) are both upper 250 triangular with \(\lambda\) along the diagonal: take \(v_1, v_2, v_3 \in 251 \mathbb{C}^3\) with \(v_1 \in E_{\alpha_2 = \lambda} \cap E_{\beta_2 = 252 \lambda}\) and \(v_2 \in V_{L_{\alpha_2}}\). Any such pair of matrices 253 satisfying the braid relation (\ref{eq:braid-rel-induction-basis}) commute. 254 255 Similarly, if \(L_{\alpha_2}\) has Jordan form (9) and \(E_{\alpha_2 = 256 \lambda} \ne E_{\beta_2 = \lambda}\) we use 257 (\ref{eq:braid-rel-induction-basis}) to conclude \(L_{\alpha_1}\) and 258 \(L_{\beta_1}\) commute -- again, see \cite[Proposition~5.1]{korkmaz}. We are 259 done. 260 \end{proof} 261 262 We are now ready to establish the triviality of low-dimensional 263 representations. 264 265 \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}] 266 Let \(g \ge 1\), \(b \ge 0\), and fix \(\rho : \Mod(\Sigma_g^b) \to 267 \GL_n(\mathbb{C})\) with \(n < 2g\). We want to show 268 \(\rho(\Mod(\Sigma_g^b))\) is Abelian. As promised, we proceed by induction 269 on \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and 270 \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also 271 established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. 272 273 Now suppose \(g \ge 3\) and every \(m\)-dimensional representation of 274 \(\Sigma_{g - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). 275 Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots, 276 \gamma_{g - 1}, \eta_1, \ldots, \eta_{b-1} \subset \Sigma_g^b\) be the curves 277 from the Lickorish generators of \(\Mod(\Sigma_g^b)\), as in 278 Figure~\ref{fig:lickorish-gens}. Once again, let \(L_\alpha = 279 \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of 280 \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(\Sigma \cong 281 \Sigma_{g - 1}^1\) be the closed subsurface highlighted in 282 Figure~\ref{fig:korkmaz-proof-subsurface}. 283 284 \begin{figure}[ht] 285 \centering 286 \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps} 287 \caption{The subsurface $\Sigma \subset \Sigma_g^b$.} 288 \label{fig:korkmaz-proof-subsurface} 289 \end{figure} 290 291 We claim that it suffices to find a \(m\)-dimensional 292 \(\Mod(\Sigma)\)-invariant\footnote{Here we view $\Mod(\Sigma)$ as a subgroup 293 of $\Mod(\Sigma_g^b)$ via the inclusion homomorphism $\Mod(\Sigma) \to 294 \Mod(\Sigma_g^b)$ from Example~\ref{ex:inclusion-morphism}, which can be 295 shown to be injective in this particular case.} subspace \(W \subset 296 \mathbb{C}^n\) with \(2 \le m \le n - 2\). Indeed, in this case \(m < 2(g - 297 1)\) and \(\dim \mfrac{\mathbb{C}^n}{W} = n - m < 2(g - 1)\). Thus both 298 representations 299 \begin{align*} 300 \rho_1 : \Mod(\Sigma) & \to \GL(W) \cong \GL_m(\mathbb{C}) 301 & 302 \rho_2 : \Mod(\Sigma) & \to \GL(\mfrac{\mathbb{C}^n}{W}) 303 \cong \GL_{n - m}(\mathbb{C}) 304 \end{align*} 305 fall into the induction hypothesis -- i.e. \(\rho_i(\Mod(\Sigma))\) is 306 Abelian. In particular, \(\rho_i(\Mod(\Sigma)') = 1\) and we can find some 307 basis for \(\mathbb{C}^n\) with respect to which 308 \[ 309 \rho(f) = 310 \left( 311 \begin{array}{c|c} 312 1_m & * \\ \hline 313 0 & 1_{n - m} 314 \end{array} 315 \right) 316 \] 317 for all \(f \in \Mod(\Sigma)'\) -- where \(1_k\) denotes the \(k \times k\) 318 identity matrix. Since the group of upper triangular matrices is solvable, it 319 follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) 320 annihilates all of \(\Mod(\Sigma)'\) and, in particular, \(\tau_{\alpha_1} 321 \tau_{\beta_1}^{-1} \in \ker \rho\). Now recall from 322 Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(\Sigma_g^b)'\) is 323 normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we 324 conclude \(\rho(\Mod(\Sigma_g^b)') = 1\), as desired. 325 326 As before, we exhaustively analyze all possible Jordan forms of 327 \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues 328 \(\lambda_1, \ldots, \lambda_k\) of \(L_{\alpha_g}\) such that the sum \(W = 329 \bigoplus_i E_{\alpha_g = \lambda_i}\) of the corresponding eigenspaces has 330 dimension \(m\) with \(2 \le m \le n - 2\). In this case, it suffices to 331 observe that since \(\alpha_g\) lies outside of \(\Sigma\), each 332 \(E_{\alpha_g = \lambda_i}\) is \(\Mod(\Sigma)\)-invariant: the Lickorish 333 generators \(\tau_{\alpha_1}, \ldots, \tau_{\alpha_{g - 1}}, \tau_{\beta_1}, 334 \ldots, \tau_{\beta_{g - 1}}\), \(\tau_{\gamma_1}, \ldots, \tau_{\gamma_{g - 335 2}}\) of \(\Sigma \cong \Sigma_{g - 1}^1\) all commute with 336 \(\tau_{\alpha_g}\) and thus preserve the eigenspaces of its action on 337 \(\mathbb{C}^n\). 338 339 If no sum of the form \(\bigoplus_i E_{\alpha_g = \lambda_i}\) has dimension 340 lying between \(2\) and \(n - 2\), then there must be at most \(2\) distinct 341 eigenvalues \(\lambda\) of \(L_{\alpha_g}\), and \(\dim E_{\alpha_g = 342 \lambda} = 1, n - 1, n\) for all such \(\lambda\). Hence the Jordan form of 343 \(L_{\alpha_g}\) has to be one of 344 \begin{align*} 345 \begin{pmatrix} 346 \lambda & 0 & 0 & \cdots & 0 & 0 \\ 347 0 & \lambda & 0 & \ldots & 0 & 0 \\ 348 \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 349 0 & 0 & 0 & \cdots & \lambda & 0 \\ 350 0 & 0 & 0 & \cdots & 0 & \lambda 351 \end{pmatrix} 352 & \quad{\normalfont(1)} 353 & 354 \begin{pmatrix} 355 \lambda & 1 & 0 & \cdots & 0 & 0 \\ 356 0 & \lambda & 1 & \ldots & 0 & 0 \\ 357 \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 358 0 & 0 & 0 & \cdots & \lambda & 1 \\ 359 0 & 0 & 0 & \cdots & 0 & \lambda 360 \end{pmatrix} 361 & \quad{\normalfont(2)} 362 \\ 363 \begin{pmatrix} 364 \lambda & 0 & 0 & \cdots & 0 & 0 \\ 365 0 & \lambda & 0 & \ldots & 0 & 0 \\ 366 \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 367 0 & 0 & 0 & \cdots & \lambda & 1 \\ 368 0 & 0 & 0 & \cdots & 0 & \lambda 369 \end{pmatrix} 370 & \quad{\normalfont(3)} 371 & 372 \begin{pmatrix} 373 \lambda & 0 & 0 & \cdots & 0 & 0 \\ 374 0 & \lambda & 0 & \ldots & 0 & 0 \\ 375 \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 376 0 & 0 & 0 & \cdots & \lambda & 0 \\ 377 0 & 0 & 0 & \cdots & 0 & \mu 378 \end{pmatrix} 379 & \quad{\normalfont(4)} 380 \end{align*} 381 for \(\lambda \ne \mu\). We analyze the first two sporadic cases 382 individually. 383 384 \begin{enumerate} 385 \item[\bfseries\color{highlight}(1)] 386 Here we use the change of coordinates principle: each \(L_{\alpha_i}, 387 L_{\beta_i}, L_{\gamma_i}, L_{\eta_i}\) is conjugate to \(L_{\alpha_g} = 388 \lambda\), so all Lickorish generators of \(\Mod(\Sigma_g^b)\) act on 389 \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. Hence 390 \(\rho(\Mod(\Sigma_g^b)) = \langle \lambda \rangle\) is Abelian. 391 392 \item[\bfseries\color{highlight}(2)] 393 In this case, \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a 394 \(2\)-dimensional \(\Mod(\Sigma)\)-invariant subspace. 395 \end{enumerate} 396 397 For cases (3) and (4), we consider two situations: \(E_{\alpha_g = \lambda} 398 \ne E_{\beta_g = \lambda}\) or \(E_{\alpha_g = \lambda} = E_{\beta_g = 399 \lambda}\). If \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\), then \(W 400 = E_{\alpha_g = \lambda} \cap E_{\beta_g = \lambda}\) is a \((n - 401 2)\)-dimensional \(\Mod(\Sigma)\)-invariant subspace: since \(\beta_g\) lies 402 outside of \(\Sigma\) and \(L_{\alpha_g}, L_{\beta_g}\) are conjugate, both 403 \(E_{\alpha_g = \lambda}\) and \(E_{\beta_g = \lambda}\) are 404 \(\Mod(\Sigma)\)-invariant \((n - 1)\)-dimensional subspaces. 405 406 Finally, we consider the case where \(E_{\alpha_g = \lambda} = E_{\beta_g 407 = \lambda}\). In this situation, as in the proof of 408 Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}, it follows from 409 Observation~\ref{ex:change-of-coordinates-crossing} that there are \(f_i, 410 g_i, h_i \in \Mod(\Sigma_g^b)\) with 411 \begin{align*} 412 f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i} 413 & 414 g_i \tau_{\alpha_g} g_i^{-1} & = \tau_{\beta_i} 415 & 416 h_i \tau_{\alpha_g} h_i^{-1} & = \tau_{\alpha_g} 417 \\ 418 f_i \tau_{\beta_g} f_i^{-1} & = \tau_{\beta_i} 419 & 420 g_i \tau_{\beta_g} g_i^{-1} & = \tau_{\gamma_i} 421 & 422 h_i \tau_{\beta_g} h_i^{-1} & = \tau_{\eta_i} 423 \end{align*} 424 and thus 425 \(E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 = 426 \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots = 427 E_{\gamma_{g - 1} = \lambda} = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b-1} 428 = \lambda}\). 429 430 In particular, we can find a basis for \(\mathbb{C}^n\) with respect to 431 which the matrix of any Lickorish generator has the form 432 \[ 433 \begin{pmatrix} 434 \lambda & 0 & \cdots & 0 & * \\ 435 0 & \lambda & \ldots & 0 & * \\ 436 \vdots & \vdots & \ddots & \vdots & \vdots \\ 437 0 & 0 & \cdots & \lambda & * \\ 438 0 & 0 & \cdots & 0 & * 439 \end{pmatrix}. 440 \] 441 Since the group of upper triangular matrices is solvable and 442 \(\Mod(\Sigma_g^b)\) is perfect, it follows that \(\rho(\Mod(\Sigma_g^b))\) is 443 trivial. This concludes the proof \(\rho(\Mod(\Sigma_g^b))\) is Abelian. 444 445 To see that \(\rho(\Mod(\Sigma_g^b)) = 1\) for \(g \ge 3\) we note that, 446 since \(\rho\) has Abelian image and thus factors though the Abelianization 447 map \(\Mod(\Sigma_g^b) \to \Mod(\Sigma_g^b)^\ab = 448 \mfrac{\Mod(\Sigma_g^b)}{[\Mod(\Sigma_g^b), \Mod(\Sigma_g^b)]}\). Now recall 449 from Proposition~\ref{thm:trivial-abelianization} that \(\Mod(\Sigma_g^b)^\ab 450 = 0\) for \(g \ge 3\). We are done. 451 \end{proof} 452 453 Having established the triviality of the low-dimensional representations \(\rho 454 : \Mod(\Sigma_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to 455 understand the \(2g\)-dimensional representations of \(\Mod(\Sigma_g^b)\). We 456 certainly know a nontrivial example of such, namely the symplectic 457 representation \(\psi : \Mod(\Sigma_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) 458 from Example~\ref{ex:symplectic-rep}. Surprisingly, this turns out to be 459 \emph{essentially} the only example of a nontrivial \(2g\)-dimensional 460 representation in the compact case. More precisely, 461 462 \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic} 463 Let \(g \ge 3\) and \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). 464 Then \(\rho\) is either trivial or conjugate to the symplectic 465 representation\footnote{Here the map $\Mod(\Sigma_g^b) \to 466 \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the 467 inclusion morphism $\Mod(\Sigma_g^b) \to \Mod(\Sigma_g)$ with the usual 468 symplectic representation $\psi : \Mod(\Sigma_g) \to 469 \operatorname{Sp}_{2g}(\mathbb{Z})$.} \(\Mod(\Sigma_g^b) \to 470 \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(\Sigma_g^b)\). 471 \end{theorem} 472 473 Unfortunately, the limited scope of these master's thesis does not allow us to 474 dive into the proof of Theorem~\ref{thm:reps-of-dim-2g-are-symplectic}. The 475 heart of this proof lies in a result about representations of the product 476 \(B_3^n = B_3 \times \cdots \times B_3\), which Korkmaz refers to as \emph{the 477 main lemma}. 478 479 \begin{lemma}[Korkmaz' main lemma]\label{thm:main-lemma} 480 Given \(i = 1, \ldots, n\), denote by 481 \begin{align*} 482 a_i & = (1, \ldots, 1, \sigma_1, 1, \ldots, 1) & 483 b_i & = (1, \ldots, 1, \sigma_2, 1, \ldots, 1) 484 \end{align*} 485 the \(n\)-tuples in \(B_3^n\) whose \(i\)-th coordinates are \(\sigma_1\) and 486 \(\sigma_2\), respectively, and with all other coordinates equal to \(1\). 487 Let \(m \ge 2n\) and \(\rho : B_3^n \to \GL_m(\mathbb{C})\) be a 488 representation satisfying: 489 \begin{enumerate} 490 \item The only eigenvalue of \(\rho(a_i)\) is \(1\) and its eigenspace is 491 \((m - 1)\)-dimensional. 492 \item The eigenspaces of \(\rho(a_i)\) and \(\rho(b_i)\) associated to the 493 eigenvalue \(1\) do not coincide. 494 \end{enumerate} 495 Then \(\rho\) is conjugate to the representation 496 \begin{align*} 497 B_3^n & \to \GL_m(\mathbb{C}) \\ 498 a_i 499 & \mapsto 500 \left( 501 \begin{array}{c|c|c} 502 1_{2(i-1)} & 0 & 0 \\ \hline 503 0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline 504 0 & 0 & 1_{m-2i} 505 \end{array} 506 \right) \\ 507 b_i 508 & \mapsto 509 \left( 510 \begin{array}{c|c|c} 511 1_{2(i-1)} & 0 & 0 \\ \hline 512 0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline 513 0 & 0 & 1_{m-2i} 514 \end{array} 515 \right), 516 \end{align*} 517 where \(1_k\) denotes the \(k \times k\) identity matrix. 518 \end{lemma} 519 520 This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice 521 that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma} 522 coincide with the action of the Lickorish generators \(\tau_{\alpha_1}, \ldots, 523 \tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(\Sigma_g^b)\) on 524 \(H_1(\Sigma_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard 525 basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for 526 \(H_1(\Sigma_g, \mathbb{C})\). 527 \begin{align*} 528 (\tau_{\alpha_i})_* & = 529 \left( 530 \begin{array}{c|c|c} 531 1 & 0 & 0 \\ \hline 532 0 & \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} & 0 \\ \hline 533 0 & 0 & 1 534 \end{array} 535 \right) & 536 (\tau_{\beta_i})_* & = 537 \left( 538 \begin{array}{c|c|c} 539 1 & 0 & 0 \\ \hline 540 0 & \begin{array}{cc} 1 & 0 \\ -1 & 1 \end{array} & 0 \\ \hline 541 0 & 0 & 1 542 \end{array} 543 \right) 544 \end{align*} 545 546 Hence by embedding \(B_3^g\) in \(\Mod(\Sigma_g^b)\) via 547 \begin{align*} 548 B_3^g & \to \Mod(\Sigma_g^b) \\ 549 a_i & \mapsto \tau_{\alpha_i} \\ 550 b_i & \mapsto \tau_{\beta_i} 551 \end{align*} 552 we can see that any \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\) in a 553 certain class of representation satisfying some technical conditions must be 554 conjugate to the symplectic representation \(\Mod(\Sigma_g^b) \to 555 \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\). 556 557 Korkmaz then goes on to show that such technical conditions are met for any 558 nontrivial \(\rho : \Mod(\Sigma_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore, 559 Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with 560 respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots, 561 \rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots, 562 \rho(\tau_{\eta_{b-1}})\) also agree with the action of \(\Mod(\Sigma_g^b)\) on 563 \(H_1(\Sigma_g, \mathbb{C})\), concluding the classification of 564 \(2g\)-dimensional representations. 565 566 Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional 567 representations of \(\Mod(\Sigma_g^b)\) for \(g \ge 7\) in terms of certain 568 twisted \(1\)-cohomology groups. On the other hand, the representations of 569 dimension \(n > 2g + 1\) are still poorly understood, and fundamental questions 570 remain unsweared. In the short and mid-terms, the works of Korkmaz and Kasahara 571 lead to many follow-up questions. For example, 572 \begin{enumerate} 573 \item In the \(g \ge 3\) case, Korkmaz \cite[Theorem~3]{korkmaz} established 574 the lower bound of \(3g - 3\) for the dimension of an injective linear 575 representation of \(\Mod(\Sigma_g)\) -- if one such representation exists. 576 Can we improve this lower bound? 577 578 \item What is the minimal dimension for a representation of 579 \(\Mod(\Sigma_g)\) which does not annihilate the entire kernel of the 580 symplectic representation \(\psi : \Mod(\Sigma_g) \to 581 \operatorname{Sp}_{2g}(\mathbb{Z})\)? In particular, do the \((2g + 582 1)\)-dimensional representations classified by Kasahara \cite{kasahara} 583 annihilate all of \(\ker \psi\)? 584 \end{enumerate} 585 586 These are some of the questions which I plan to work on during my upcoming PhD.